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We study dynamic heterogeneities in the out-of-equilibrium coarseningdynamics of the spherical ferromagnet after a quench from infinite temperatureto its critical point. A standard way of probing such heterogeneities is bymonitoring the fluctuations of correlation and susceptibility, coarse-grainedover mesoscopic regions. We discuss how to define fluctuating coarse-grainedcorrelations (C) and susceptibilities (Chi) in models where no quencheddisorder is present. Our focus for the spherical model is on coarse-grainingover the whole volume of $N$ spins, which requires accounting for N^{-1/2}non-Gaussian fluctuations of the spin. The latter are treated as a perturbationabout the leading order Gaussian statistics. We obtain exact results for thesequantities, which enable us to characterise the joint distribution of C and Chifluctuations. We find that this distribution is qualitatively different, evenfor equilibrium above criticality, from the spin-glass scenario where C and Chifluctuations are linked in a manner akin to the fluctuation-dissipationrelation between the average C and Chi. Our results show that coarsening atcriticality is clearly heterogeneous for d>4 and suggest that, as in otherglassy systems, there is a well-defined timescale on which fluctuations acrossthermal histories are largest. Surprisingly, however, neither this timescalenor the amplitude of the heterogeneities increase with the age of the system,as would be expected from the growing correlation length. For d<4, the strengthof the fluctuations varies on a timescale proportional to the age of thesystem; the corresponding amplitude also grows with age, but does not scalewith the correlation volume as might have been expected naively.

Dynamic heterogeneities in critical coarsening: Exact results for correlation and response fluctuations in finite-sized spherical models